Optimal Control of Flood Diversion in Watershed Using Nonlinear Optimization

Optimal Control of Flood Diversion in Watershed Optimal Control of Flood Diversion in Watershed

Using Nonlinear OptimizationUsing Nonlinear Optimization

National Center for Computational Hydroscience and EngineeringThe University of Mississippi

Yan Ding, Ph.D.1 and Sam S. Y. Wang, Ph.D. P.E.2

1Research Assistant Professor, 2 F.A.P. Barnard Distinguished Professor and Director

National Center for Computational Hydroscience and Engingeerin, The University of Mississippi, Oxford, MS 38677

Presented at Conference of 50 Years of Soil and Water Research In a Changing Agricultural Environment, Oxford, MS, Sept 4, 2008

OutlineOutline

IntroductionIntroduction

Nonlinear Models for Forecasting Flood EventsNonlinear Models for Forecasting Flood Events

Nonlinear Optimization Scheme for Finding the Nonlinear Optimization Scheme for Finding the Optimal Flood Diversion Hydrograph to Mitigate Optimal Flood Diversion Hydrograph to Mitigate Hazardous Storm WatersHazardous Storm Waters

Applications to a Variety of Flood Diversion Control Applications to a Variety of Flood Diversion Control ScenariosScenarios

Conclusions and Future Research Topics Conclusions and Future Research Topics

The spillway (highlighted in green) stretches from the Mississippi River, at right, northward to Lake Ponchartrain, on the left of the photo.

An Example of Flood DiversionAn Example of Flood Diversion – The Bonnet Carre’ Spillway – The Bonnet Carre’ Spillway

Flooded Street, Mississippi River Flood of 1927

The Bonnet Carré Spillway, the southern-most floodway in the Mississippi River and Tributaries system, has historically been the first floodway in the Lower Mississippi River Valley opened during floods. The USACE’s hydraulic engineers rely on discharge and gauge readings at Red River Landing, about 200 miles above New Orleans, to determine when to open the spillway. The discharge takes two days to reach the city from the landing. As flows increase, bays are opened at Bonnet Carré to divert them.

Difficulties in Optimal Control of Open Channel Flow Difficulties in Optimal Control of Open Channel Flow

Temporally/spatially non-uniform open channel flowTemporally/spatially non-uniform open channel flow Requires that a forecasting model can predict accurately complex water Requires that a forecasting model can predict accurately complex water

flows in space and time in single channel and channel networkflows in space and time in single channel and channel network

Nonlinearity of flow controlNonlinearity of flow control Nonlinear process control, Nonlinear optimizationNonlinear process control, Nonlinear optimization Difficulties to establish the relationship between control actions and Difficulties to establish the relationship between control actions and

responses of the hydrodynamic variablesresponses of the hydrodynamic variables

Requirement of fast flow solver and optimization Requirement of fast flow solver and optimization In case of fast propagation of flood wave, a very short time is available for In case of fast propagation of flood wave, a very short time is available for

predicting the flood flow at downstream. Due to the limited time for predicting the flood flow at downstream. Due to the limited time for making decision of flood mitigation, it is crucial for decision makers to making decision of flood mitigation, it is crucial for decision makers to have a very efficient forecasting model and a control model.have a very efficient forecasting model and a control model.

Objectives

Theoretically, • Through adjoint sensitivity analysis, make nonlinear optimization

capable of flow control in complex channel shape and channel network in watershed

Real-Time Nonlinear Adaptive Control Applicable to unsteady river flows

• Establish a general numerical model for controlling hazardous floods so as to make it applicable to a variety of control scenarios

Flexible Control System; and a general tool for real-time flow control

For Engineering Applications,• Integrate the control model with the CCHE1D flow model,

• Apply to practical problems

Integrated Watershed & Channel Network Modeling with Integrated Watershed & Channel Network Modeling with CCHE1DCCHE1D

Digital ElevationModel (DEM)

Rainfall-Runoff Simulation

Upland Soil Erosion(AGNPS or SWAT)

Channel Network Flow and Sediment Routing

(CCHE1D)

Channel Network andSub-basin Definition

(TOPAZ)

q(t)

=?

01

qx

Q

t

AL

02 2

2

2

fgSx

Zg

A

Q

xA

Q

t

L

3/42

2 ||

RA

QQnS f

Dynamic Wave ModelDynamic Wave Modelfor Flood Wave Predictionfor Flood Wave Prediction

A=Cross-sectional Area; q=Lateral outflow;=correction factor; R=hydraulic radius n = Manning’s roughness

where Q = discharge; Z=water stage;

• Boundary Conditions

• Initial Conditions (Base Flows)

• Internal Flow Conditions for Channel Network

A Typical Hydrograph by USGS

Control Actions Control Actions - Available Control Variables in Open Channel Flow- Available Control Variables in Open Channel Flow

Control lateral flow at a certain location Control lateral flow at a certain location xx00: Real-time flow : Real-time flow diversion rate diversion rate q(xq(x00, t), t) at a spillwayat a spillway

Control lateral flow at the optimal location Control lateral flow at the optimal location xx: Real-time levee : Real-time levee breaching rate breaching rate q(x, t)q(x, t) at the optimal locationat the optimal location

Control upstream discharge Control upstream discharge Q(0, t)Q(0, t):: real-time reservoir release real-time reservoir release

Control downstream stage Control downstream stage Z(L, t)Z(L, t):: real-time gate operation real-time gate operation

Control downstream discharge Control downstream discharge Q(L, t)Q(L, t):: real-time pump rate real-time pump rate controlcontrol

Control bed friction (roughness Control bed friction (roughness nn): ):

+2.0m

+0.0m

20m

70m

Zobj

An Objective Function for Flood ControlAn Objective Function for Flood Control

To evaluate the discrepancy between predicted and maximum To evaluate the discrepancy between predicted and maximum allowable stages, a weighted form is defined asallowable stages, a weighted form is defined as

where T=control duration; L = channel length; t=time; x=distance along channel; Z=predicted water stage; Zobj(x) =maximum allowable water stage in river bank (levee) (or objective water stage); x0= target location where the water stage is protective; = Dirac delta function

)()(,0

)()(),()](),([

00

0004

xZxZif

xZxZifxxxZtxZLT

Wr

obj

objobjZ

dxdttxqQZrJT L

0 0

),,,,(

Mathematical Framework for Optimal ControlMathematical Framework for Optimal Control

The optimazition is to find the control variable The optimazition is to find the control variable qq satisfying satisfying a dynamic system such thata dynamic system such that

where where QQ and and ZZ are satisfied with the continuity equation are satisfied with the continuity equation and momentum equation, respectively (i.e., de Saint and momentum equation, respectively (i.e., de Saint Venant Equations)Venant Equations)

Local minimum theory :Local minimum theory : Necessary ConditionNecessary Condition: If : If nn** is the true value, then is the true value, then J(nJ(n**)=0)=0;; Sufficient ConditionSufficient Condition: If the Hessian matrix : If the Hessian matrix 22J(nJ(n**) is ) is

positive definitepositive definite, then , then nn** is a strict local minimizer of is a strict local minimizer of ff

)),,,(min()( qZQJqf

Sensitivity AnalysisSensitivity Analysis- - Establishing A Relationship between Control Actions and System VariablesEstablishing A Relationship between Control Actions and System Variables

Compute the gradient of objective function, Compute the gradient of objective function, qq((XX, q),, q), i.e., sensitivity of i.e., sensitivity of control variable throughcontrol variable through

1. 1. Influence Coefficient MethodInfluence Coefficient Method (Yeh, 1986):(Yeh, 1986): Parameter perturbation trial-and-error; lower accuracyParameter perturbation trial-and-error; lower accuracy

2. 2. Sensitivity Equation Method Sensitivity Equation Method (Ding, Jia, & Wang, 2004)(Ding, Jia, & Wang, 2004)

Directly compute the sensitivity Directly compute the sensitivity ∂X/∂q∂X/∂q by solving the sensitivity equations by solving the sensitivity equations Drawback: different control variables have different forms in the equations, no Drawback: different control variables have different forms in the equations, no

general measures for system perturbations; The number of sensitivity equations = the general measures for system perturbations; The number of sensitivity equations = the number of control variables.number of control variables.

Merit: Forward computation, no worry about the storage of codesMerit: Forward computation, no worry about the storage of codes

3. 3. Adjoint Sensitivity Method Adjoint Sensitivity Method (Ding and Wang, 2003)(Ding and Wang, 2003)

Solve the governing equations and their associated adjoint equations sequentially. Solve the governing equations and their associated adjoint equations sequentially. Merit: general measures for sensitivity, limited number of the adjoint equations Merit: general measures for sensitivity, limited number of the adjoint equations

(=number of the governing equations) regardless of the number of control variables.(=number of the governing equations) regardless of the number of control variables. Drawback: Backward computation, has to save the time histories of physical variables Drawback: Backward computation, has to save the time histories of physical variables

before the computation of the adjoint equations.before the computation of the adjoint equations.

x

t

A B

CD

O L

T

Variational AnalysisVariational Analysis- - To Obtain Adjoint EquationsTo Obtain Adjoint Equations

Extended Objective Function

dxdtLLJJT L

QA 0 0 21

* )(

where A and Q are the Lagrangian multipliers

Fig. Solution domain

Necessary Condition

0* JJ on the conditions that

0),(0),(

1

2{

ZQLZQL

Variation of Extended Objective FunctionVariation of Extended Objective Function

0

*

])([])[(

)(

||2

)||21

(

)||)1(2

(

)(

23

2

2

0 0

0 0 2

0 0 22

0 0 3

3/2

3

2

2*

0 0

dtQA

QA

A

QdxQ

AA

A

Q

qdxdt

ndxdtnK

QgQ

QdxdtK

Qg

xA

Q

tAx

AdxdtnK

QQRg

xA

Q

tA

Q

xB

g

t

dxdtqq

rQ

Q

rδA

A

Z

Z

r

QAQQ

QA

T L

A

T L Q

T L QQQA

T L QQQQA

T LJ

where

*;3

42;

*

*3/2

BB

RB

n

ARK Top width of channel

Variations of Variations of JJ with Respect to Control Variables with Respect to Control Variables – Formulations of Sensitivities– Formulations of Sensitivities

dttQQ

rtOQJ

T

x

A ),0()()),((0

0

ndxdtnK

QgQ

n

rnJ

T L Q

0 0 2

||2)(

dxdttxqq

rtxqJ

T L

A ),()),((0 0

dttLAA

Q

B

g

A

rtLAJ

Lx

T

Q ),()()),((0 3

2

*

Lateral Outflow

Upstream Discharge

Downstream Section Area or Stage

Bed Friction

Remarks: Control actions for open channel flows may rely on one control variable or a rational combination of these variables. Therefore, a variety of control scenarios principally can be integrated into a general control model of open channel flow.

General Formulations of Adjoint Equations for General Formulations of Adjoint Equations for the Full Nonlinear Saint Venant Equationsthe Full Nonlinear Saint Venant Equations

Q

r

A

Q

A

r

AK

QQg

xB

g

xA

Q

tQQAA

2*

||)1(2

Q

r

K

QgA

xA

xA

Q

t QAQQ

2

2

According to the extremum condition, all terms multiplied by A and Q can be set to zero, respectively, so as to obtain the equations of the two Lagrangian multipliers, i.e, adjoint equations (Ding & Wang 2003)

Transversality Conditions and Boundary Transversality Conditions and Boundary ConditionsConditions

x

t

A B

CD

O L

T

Fig. Solution domain

Considering the contour integral in J*, This term I needs to be zero.

0

])()[(])[(23

2

*2

DACDBCAB

QAQQ

QA dtQA

QA

A

Q

B

gdxQ

AA

A

QI

],0[,0),(

],0[,0),(

LxTx

LxTx

A

Q

Transversality (Final) Conditions

],0[,0),0( TttQ

],0[),,(),(2

TttLQ

AtL AQ

Upstream B.C.

Downstream B.C.

Backward Computation

Internal Boundary Conditions Internal Boundary Conditions – – for Channel Networkfor Channel Network

213

321

QQQ

ZZZ

I.B.C.s of Adjoint Equations

32

22

12

A

Q

A

Q

A

Q QA

QA

QA

I.B.C.s of Flow Model

Fig. Confluence

3

3

2*

2

3

2*

1

3

2*

xx

Q

xx

Q

xx

Q A

QBg

A

QBg

A

QBg

Numerical TechniquesNumerical Techniques

])1()[1(])1([),( 111

1ni

ni

ni

nitx

ttt

tx ni

ni

ni

ni

1

11

1 )1(),(

xxx

tx ni

ni

ni

ni

111

1 )1(),(

1-D Time-Space Discretization (Preissmann, 1961)

Solver of the resulting linear algebraic equations (Pentadiagonal Matrix)

Double Sweep Algorithm based on the Gauss Elimination

where and are two weighting parameters in time and space, respectively;t=time increment; x=spatial length

Minimization Procedures for Nonlinear Minimization Procedures for Nonlinear OptimizationOptimization

CG MethodCG Method ( (Fletcher-Reeves methodFletcher-Reeves method) () (Fletcher 1987Fletcher 1987)) The convergence direction of minimization is considered as The convergence direction of minimization is considered as

the gradient of objective functionthe gradient of objective function.. Trust Region MethodTrust Region Method (e.g (e.g Sakawa-Shindo methodSakawa-Shindo method)) considering the considering the first order derivativefirst order derivative of performance function of performance function

only, stable in most of practical problems (only, stable in most of practical problems (Ding et al 2004Ding et al 2004)) Limited-Memory Quasi-Newton MethodLimited-Memory Quasi-Newton Method (LMQN) (LMQN) Newton-like method, applicable for large-scale computation, Newton-like method, applicable for large-scale computation,

considering the considering the second order derivativesecond order derivative of objective function of objective function (the approximate Hessian matrix) (Ding & Wang 2005)(the approximate Hessian matrix) (Ding & Wang 2005)

OthersOthers

Minimization ProceduresMinimization Procedures

Limited-Memory Quasi-Newton Method (LMQN)Limited-Memory Quasi-Newton Method (LMQN) Newton-like method, applicable for large-scale computation Newton-like method, applicable for large-scale computation

(with a large number of control parameters), considering the (with a large number of control parameters), considering the second order derivativesecond order derivative of objective function (the approximate of objective function (the approximate Hessian matrix)Hessian matrix)

AlgorithmsAlgorithms::

BFGS (named after its inventors, BFGS (named after its inventors, BBroyden, royden, FFletcher, letcher, GGoldfarb, and oldfarb, and SShanno)hanno)

L-BFGSL-BFGS (unconstrained optimization) (unconstrained optimization)

L-BFGS-BL-BFGS-B (bound constrained optimization) (bound constrained optimization)

Limited-Memory Quasi-Newton Method (LMQN)Limited-Memory Quasi-Newton Method (LMQN) (Basic Concept 1) (Basic Concept 1)

Given the iteration of a line search method for parameter Given the iteration of a line search method for parameter qq

qqk+1k+1 = = qqkk + + kkddkk

kk = the step length of line search = the step length of line search

sufficient decrease and curvature conditions sufficient decrease and curvature conditions

ddkk = the search direction (descent direction) = the search direction (descent direction)

BBkk = = nnnn symmetric positive definite matrixsymmetric positive definite matrix

For For the Steepest Descent Methodthe Steepest Descent Method: : BBkk = I = I

Newton’s MethodNewton’s Method: : BBkk== 22J(nJ(nkk))

Quasi-Newton MethodQuasi-Newton Method: :

BBkk= an approximation of the Hessian = an approximation of the Hessian 22J(nJ(nkk))

)(1kkkk nJBd

.

qi

qj

.q* d1

Contour of J

Flow chart of Finding optimal control variable by using LMQN procedure

Set the initial q

k=0

Solve the initial state vector X0 Flow Model (CCHE1D)

Calculation of objective function J0, gradient g0, and search

direction d0

Calculation of )( 11 kk qJg

||gk+1||max{1,||qk+1||}

Calculation of Jk+1

Stop

Yes

No

Calculate kkkk dqq 1 Line Search

Solver of Adjoint Equations

Calculation of 111 kkk gHd

Update Hessian matrix by the recursive iteration

nlk

lk

lk

q

qqMax

)( 1 Yes

Yes

Solve the state vector Xk+1

L-BFGS

JkJ 1

Flow Model (CCHE1D)

Solver of Adjoint Equations

Three Major Modules• Flow Solver• Sensitivity Solver• Minimization Process

L-BFGS-BL-BFGS-B

The purpose of the L-BFGS-B method is to The purpose of the L-BFGS-B method is to minimize the objective function minimize the objective function J(q)J(q) , i.e., , i.e.,

min J(q),min J(q),subject to the following simple bound constraint,subject to the following simple bound constraint,

qqminmin q q q qmaxmax,,

where the vectors where the vectors qqminmin and and qqmaxmax mean lower and upper mean lower and upper bounds on the control variables.bounds on the control variables.

L-BFGS-B is an extension of the limited memory L-BFGS-B is an extension of the limited memory algorithm (L-BFGS) (algorithm (L-BFGS) (Liu & Nocedal, 1989) Liu & Nocedal, 1989) for bound for bound constrained optimization (constrained optimization (Byrd et al, 1995)Byrd et al, 1995) . .

Flooding and Flood ControlFlooding and Flood Control

Levee Failure, 1993 flood. Missouri. Flood Gate, West Atchafalaya Basin, Charenton Floodgate, Louisiana

Control of Flood Diversion in A Single Channel Control of Flood Diversion in A Single Channel – A Simplified Problem– A Simplified Problem

q(xc,t) = ?

xc

No Control

Zobj(x0,t)

Under Control

Z(x0,t) A Tolerable Stage

t

Objective Function

dxdttxqQZfLT

JT L

0 0

),,,,(1

obj

objobjZ

ZZif

ZZifxxxZtxZWf

,0

),()](),([ 04

Optimal Control of Flood Diversion Rate Optimal Control of Flood Diversion Rate ( Case 1) -( Case 1) - A Hypothetic Single Channel A Hypothetic Single Channel

Time

Dis

char

ge

TpTd

Qp

Qb

+2.0m

+0.0m

20m

70m

1:2

1:1.

5

A Triangular HydrographCross-section

Parameter L x t n QP Qb Tp Td Z0 Wz

Unit (km) (km) (min) s/m1/3 (m3/s) (m3/s) (hour) (hour) m

Value 10.0 0.5 5.0 1.0(0.55*) 0.5 0.03 100.0 10.0 16.0 48.0 3.5 103

* This value is used for solving adjoint equations

Lateral Outflow

Z0=3.5m

Optimal Lateral Outflow and Objective Function Optimal Lateral Outflow and Objective Function (Case 1)(Case 1)

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-100

-50

0

50

100

Iteration= 1Iteration= 3Iteration= 4Iteration= 5Iteration= 6Iteration= 10Iteration= 30Iteration= 70

Hydrograph at inlet

Iterations of L-BFGS-BO

bjec

tive

Fun

ctio

n

Nor

mof

Gra

dien

t

0 10 20 30 40 50 60 7010-3

10-2

10-1

100

101

102

103

104

105

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Objective FunctionNorm of Gradient

Iterations of optimal lateral outflowObjective function and Norm of gradient of the function

Optimal Outflow q

Comparison of Water Stages in Space and Time Comparison of Water Stages in Space and Time (Case 1)(Case 1)

Km

01

23

45

67

89

10Hours

012

2436

48

Wat

erS

tage

(m)

0

1

2

3

4

5

No Control Optimal Control of Lateral Outflow

Km

01

23

45

67

89

10Hours

012

2436

48

Wat

erS

tage

(m)

0

1

2

3

4

5

Lateral Outflow

Allowable Stage Z0=3.5

Comparison of Discharge in Time and Space Comparison of Discharge in Time and Space (Case 1)(Case 1)

Km

0 1 2 3 4 5 6 7 8 9 10 Hours0

1224

3648

Dis

cha

rge

(m3/s

)

20

40

60

80

100

Lateral Outflow

Km

0 1 2 3 4 5 6 7 8 9 10 Hours0

1224

3648

Dis

char

ge(m

3 /s)

20

40

60

80

100

No Control Optimal Control of Lateral Outflow

Sensitivity Sensitivity ∂J/∂q(x,t)∂J/∂q(x,t)

Hours

A0 10 20 30 40 50

0

5E-06

1E-05

1.5E-05

2E-05

2.5E-05ITERATION= 1ITERATION= 3ITERATION= 4ITERATION= 5ITERATION= 6ITERATION= 10ITERATION= 30

Km

01

23

45

67

89

10 Hours

012

2436

48

A0

2E-05

4E-05

Lateral Discharge

Sensitivity of q in time and space at the 1st iteration

Iterative history of sensitivity at the control point

Fast searching

Optimal Control of Lateral Outflow (Case 2) Optimal Control of Lateral Outflow (Case 2) –Under the limitation of the maximum lateral outflow rate–Under the limitation of the maximum lateral outflow rate

Suppose that the maximum lateral outflow rate is specified due to the limited capacity of flood gate or pump station, e.g. q 50.0 m3/s

Bound Constraints:

Application of the quasi-Newton method with bound constraints (L-BFGS-B)

Lateral Outflow q≤q0

Z0=-3.5m

Optimal Lateral Outflow with ConstraintOptimal Lateral Outflow with Constraint

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-100

-50

0

50

100

Iteration= 1Iteration= 3Iteration= 4Iteration= 5Iteration= 6Iteration= 10Iteration= 30Iteration= 70

Hydrograph at inlet

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-100

-50

0

50

100

Case 1Case 2

Hydrograph at inlet

Iterations of optimal lateral outflow Comparison of optimal lateral outflow rates between Case 1 and Case 2

Controlled Stage and Discharge in the Channel Controlled Stage and Discharge in the Channel (Case 2)(Case 2)

Km

01

23

45

67

89

10Hours

012

2436

48 Wat

erS

tage

(m)

0

1

2

3

4

5

Lateral Outflow

Km

0 1 2 3 4 5 6 7 8 9 10 Hours0

1224

3648

Dis

char

ge(m

3 /s)

20

40

60

80

100

Lateral Outflow

Stage in time and space Discharge in time and space

Allowable stage Z0=3.5m

Optimal Control of Lateral Outflows Optimal Control of Lateral Outflows – Multiple Lateral Outflows (Case 3)– Multiple Lateral Outflows (Case 3)

Suppose that there are three flood gates (or spillways) in upstream, middle reach, and downstream.

Condition of control:

Z0=3.5m

q1 q2q3

Optimal Lateral Outflow Rates in Three Diversions Optimal Lateral Outflow Rates in Three Diversions (Case 3)(Case 3)

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-100

-50

0

50

100

q1

q2

q3

Hydrograph at inlet

HoursD

isch

arge

(m3 /s

)0 10 20 30 40 50

-100

-50

0

50

100

Lateral Outflow

Hydrograph at inlet

Optimal lateral outflow rates of three floodgates (Case 4)

Optimal lateral outflow of only one gate (=q1) (Case 1)

Controlled Stage and Discharge by Three Diversions Controlled Stage and Discharge by Three Diversions (Case 3)(Case 3)

Km

01

23

45

67

89

10Hours

012

2436

48

Wat

erS

tage

(m)

0

1

2

3

4

5

q 1

q 2

q 3 Km

01

23

45

67

89

10 Hours0

1224

3648

Dis

char

ge(m

3 /s)

20

40

60

80

100

q 3Lateral Outflo

w: q 1

q 2

Stage in time and space Discharge in time and space

Allowable stage Z0=3.5m

Comparisons of Diversion Percentages and Comparisons of Diversion Percentages and Objective Functions Objective Functions

Case Case qqmaxmax Number of Number of floodgatefloodgate

11 N/AN/A 11

22 50.050.0mm33/s/s 11

33 N/AN/A 33

Iterations of L-BFGS-B

Obj

ectiv

eF

unct

ion

0 10 20 30 40 50 60 7010-7

10-5

10-3

10-1

101

103

Case 1Case 2Case 43

CaseCase DiversionDiversion Volume Volume

(m(m33))

Percentage ofPercentage of Diversion Diversion

(%)(%)

11 3,952,2313,952,231 41.341.3

22 3,743,3793,743,379 39.139.1

33 3,180,6613,180,661 33.233.2

Control of Flood Diversion in A Channel NetworkControl of Flood Diversion in A Channel Network

L3 = 13,000m

L2 = 4,500m

L1

=4,

000m

1

2

3

Channel No.

Optimal Control of One Lateral Outflow in a Channel Optimal Control of One Lateral Outflow in a Channel Network (Case 5) Network (Case 5)

Channel No.

QP (m3/s)

Qb (m3/s)

Tp (hour)

Td (hour)

Z0 (m)

1 50.0 2.0 16.0 48.0 3.5 2 50.0 2.0 16.0 48.0 3.5 3 60.0 6.0 16.0 48.0 3.5

+2.0m

+0.0m

20m

70m

1:2

1:1.

5

Z0=3.5m

q(t)=?

Compound Channel Section

Time

Dis

char

ge

TpTd

Qp

Qb

Time

Dis

char

ge

TpTd

Qp

Qb

Time

Dis

char

ge

TpTd

Qp

Qb

Confluence

Optimal Lateral Outflow and Objective Function Optimal Lateral Outflow and Objective Function (Case 5: Channel Network)(Case 5: Channel Network)

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-150

-100

-50

0

50

Iteration= 1Iteration= 3Iteration= 4Iteration= 6Iteration= 10Iteration= 30Iteration= 70

Hydrograph at inlet of main stem

Tp

Hydrograph at two branchs


Obj

ectiv

eF

unct

ion

Nor

mof

Gra

dien

t

0 10 20 30 40 50 60 7010-3

10-2

10-1

100

101

102

103

104

105

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Objective FunctionNorm of Gradient

L3 =13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5

No ControlOptimal Control

Allowable Stage

Comparisons of Stages (Case 5)Comparisons of Stages (Case 5)

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

L3 =13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Comparisons of Discharges (Case 5)Comparisons of Discharges (Case 5)

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100

150No ControlOptimal Control

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100


Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100


Discharge increased !!

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100

150


Discharge increased !!

L3 = 13,000m

L2 = 4,500m

L1

=4,

000m

1

2

3

Channel No.

Optimal Control of Multiple Lateral Outflows in a Optimal Control of Multiple Lateral Outflows in a Channel Network (Case 6) Channel Network (Case 6)

Channel No.

QP (m3/s)

Qb (m3/s)

Tp (hour)

Td (hour)

Z0 (m)

1 50.0 2.0 16.0 48.0 3.5 2 50.0 2.0 16.0 48.0 3.5 3 60.0 6.0 16.0 48.0 3.5

+2.0m

+0.0m

20m

70m

1:2

1:1.

5

Z0=3.5m

q3(t)=?

Compound Channel Section

Time

Dis

char

ge

TpTd

Qp

Qb

Time

Dis

char

ge

TpTd

Qp

Qb

Time

Dis

char

ge

TpTd

Qp

Qb

q2(t)=?

q1(t)=?

Optimal Lateral Outflow Rates and Objective FunctionOptimal Lateral Outflow Rates and Objective Function (Case 6) (Case 6)

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 50-150

-100

-50

0

50

q1

q2

q3

Hydrograph at inlet of main stem

Tp

Hydrograph at two branchs


Obj

ectiv

eF

unct

ion

0 20 40 60 80 100

10-6

10-4

10-2

100

102

Case 5Case 6

Optimal lateral outflow rates at three diversions

Comparison of objective function

One Diversion

Three Diversions

Comparisons of Stages (Case 6)Comparisons of Stages (Case 6)

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

Hours

Wat

erS

tage

(m)

0 10 20 30 40 500

1

2

3

4

5


Allowable Stage

L3 =13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Comparisons of Discharges (Case 6)Comparisons of Discharges (Case 6)

L3 =13,000m

L2 = 4,500m

L 1=

4,00

0m

1

2

3

Channel No.

Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100


Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100


Hours

Dis

char

ge(m

3 /s)

0 10 20 30 40 500

50

100


Flood Diversion Control in River Flow (Real Storms)Flood Diversion Control in River Flow (Real Storms)

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Allowable Elevations along the River and Rating Allowable Elevations along the River and Rating Curve at OutletCurve at Outlet

Allowable Elevations at Cross Sections

4

5

6

7

8

9

10

0 500 1000 1500 2000 2500 3000 3500

X (m)

Ele

vati

on

(m

)

Maximum Bank Elevation (m)

Minimum Elevation (m)

Allowable Elevation (m)

Rating Curve at Outlet

4

4.5

5

5.5

6

6.5

7

7.5

0 5 10 15 20 25 30 35 40 45 50

Discharge (m3/s)

Wa

ter

Ele

vat

ion

(m

)

Rating Curve by Regression

Measured Data

Zobj (x) Z-Q

Optimal Control of One Flood Gate in River FlowOptimal Control of One Flood Gate in River Flow

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

Optimal diversion hydrograph

Storm Hydrograph

Comparison of Stages

Comparisons of Water StagesComparisons of Water Stages

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 305.4

5.6

5.8

6

6.2

6.4

6.6

6.8

Stage without controlControlled stage

Allowable stage

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 306.4

6.6

6.8

7

7.2

7.4

7.6

7.8

8

8.2


Allowable stage

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 307.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9


Allowable stage

Comparisons of DischargesComparisons of Discharges

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Discharge without controlControlled discharge

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40


Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40


Optimal Control of Two Floodgates in River FlowOptimal Control of Two Floodgates in River Flow

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Comparisons of Water Stages (Two Floodgates)Comparisons of Water Stages (Two Floodgates)

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 305.4

5.6

5.8

6

6.2

6.4

6.6

6.8


Allowable stage

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 306.4

6.6

6.8

7

7.2

7.4

7.6

7.8

8

8.2


Allowable stage

Days

Wa

ter

Sta

ge

(m)

0 5 10 15 20 25 307.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9


Allowable stage

Comparisons of Discharges (Two Floodgates)Comparisons of Discharges (Two Floodgates)

Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40


Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40


Days

Dis

cha

rge

(m3/s

)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40


Comparison of Objective FunctionsComparison of Objective Functions


Ob

ject

ive

Fu

nct

ion

0 10 20 30 40 5010-7

10-5

10-3

10-1

101

103

Control of One FloodgateControl of Two Floodgates

Data Flows for Optimal Control Based on the Data Flows for Optimal Control Based on the CCHE1D Flow ModelCCHE1D Flow Model

Model of Optimal Flow Control Based on

the CCHE1D

Input data for the CCHE1D, e.g., *.bc, *.bf

Objective data: Filename: case.obs

Initial control variable data Filename: case.cnt

Control data of L-BFGS-B: Filename: case.lbf

Output data from the CCHE1D

Results of control variables: Filename: case.par iterate.dat

Results of objective Function: Filename: case.per

History output at every nodal point: case_long.plt

Input Data Output Data

ConclusionsConclusions

The Adjoint Sensitivity Analysis provides the nonlinear flow control with The Adjoint Sensitivity Analysis provides the nonlinear flow control with comprehensive and accurate measures of sensitivities on control actions.comprehensive and accurate measures of sensitivities on control actions.

The control model is capable of solving a large-scale flow control problem The control model is capable of solving a large-scale flow control problem efficiently.efficiently.

The integrated flow model (the CCHE1D) and the adjoint equations are suitable The integrated flow model (the CCHE1D) and the adjoint equations are suitable for computing channel network with complex geometries; By taking the for computing channel network with complex geometries; By taking the advantages of the flow model in dealing with channel network, this control advantages of the flow model in dealing with channel network, this control model can be applied readily to realistic flow control problems in natural model can be applied readily to realistic flow control problems in natural streams and channel network. streams and channel network.

The adaptive control framework is general and available for practicing a variety The adaptive control framework is general and available for practicing a variety of flow control actions in open channel, e.g., flood diversion, damgate of flow control actions in open channel, e.g., flood diversion, damgate operation, and water delivery.operation, and water delivery.

The control model also can assist engineers to plan the best locations and The control model also can assist engineers to plan the best locations and capacities of floodgates from hydrodynamic point of view.capacities of floodgates from hydrodynamic point of view.

Research Topics In the FutureResearch Topics In the Future

Control of Flood and Bed ChangesControl of Flood and Bed Changes Find a real case to apply the model to flood control Find a real case to apply the model to flood control

problem or water delivery problem;problem or water delivery problem; Flood control with water security management;Flood control with water security management; Develop further modules for other process controls, Develop further modules for other process controls,

e.g. water disposal control, water quality control, e.g. water disposal control, water quality control, sediment transport and morphological process control;sediment transport and morphological process control;

Flow controls with uncertainties under natural Flow controls with uncertainties under natural conditionsconditions

OthersOthers

AcknowledgementsAcknowledgements

This work was a result of research sponsored by the This work was a result of research sponsored by the USDA Agriculture Research Service under Specific USDA Agriculture Research Service under Specific Research Agreement No. 58-6408-2-0062 (monitored Research Agreement No. 58-6408-2-0062 (monitored by the USDA-ARS National Sedimentation by the USDA-ARS National Sedimentation Laboratory) and The University of Mississippi. Laboratory) and The University of Mississippi.

Special appreciation is expressed to Dr. Sam S. Y. Special appreciation is expressed to Dr. Sam S. Y. Wang, Dr. Mustafa Altinakar, Dr. Weiming Wu, and Wang, Dr. Mustafa Altinakar, Dr. Weiming Wu, and Dr. Dalmo Vieira for their comments and Dr. Dalmo Vieira for their comments and cooperation.cooperation.

Documents

Optimal Control of Flood Diversion in Watershed Using Nonlinear Optimization